Schrodinger equation molecular orbital

Why does symmetry matter Why must there be a conservation of orbital symmetry Recall from Chapter 14 that molecular orbitals are just convenient creations of an approximation to the Schrodinger equation. Yet, orbital wavefunctions do have inherent symmetries. Wavefunctions of different symmetry cannot mix, or stated more accurately, their mixture does not give any net interaction. As we will explain further below, it is fruitless to mix them. To see this better, we now turn to an analysis of the mixture of total wavefunctions, where symmetry again is of paramount importance. Here, the theoretical basis for why symmetry matters is more clear. 

Much of quantum chemistry attempts to make more quantitative these aspects of chemists view of the periodic table and of atomic valence and structure. By starting from first principles and treating atomic and molecular states as solutions of a so-called Schrodinger equation, quantum chemistry seeks to determine what underlies the empirical quantum numbers, orbitals, the aufbau principle and the concept of valence used by spectroscopists and chemists, in some cases, even prior to the advent of quantum mechanics. 

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact solutions of the Schrodinger equation for the hydrogen atom (or any one-electron atom, such as Li" ). Hyper-Chem uses Slater atomic orbitals to construct semi-empirical molecular orbitals. The complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and are not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

Now that you know the mathematical form, you can solve the independent-electron Schrodinger equation for the molecular orbitals. First substitute the LCAO form above into equation (47) on page 193, multiply on the left by and integrate to represent... 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs " " in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Use of high-level ab initio molecular orbital methods or density functional Hamiltonians to simulate the QM region through approximate solutions of the electronic Schrodinger equation. 

These equations, derived from the Schrodinger equation of Quantum Mechanics, can be solved iteratively for matrices and jL, containing as elements the appropriately normalized molecular orbital (MO) coefficients and orbital energy eigenvalues of eq. 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

One of the more radical approximations introduced in the deduction of the Hartree-Fock equations 2 from the Schrodinger equation 3 is the assumption that the wavefunction can be expressed as a single Slater determinant, an antisymmetrized product of molecular orbitals. This is not exact, because the correct wavefunction is in fact a linear combination of Slater determinants, as shown in equation 5, where Di are Slater determinants and c are the coefficients indicating their relative weight in the wavefunction. 

The second class of theories can be characterized as attempts to find approximate solutions to the Schrodinger equation of the molecular complex as a whole. Two approaches became important in numerical calculations perturbation theory (PT) and molecular orbital (MO) methods. 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

Molecular orbital an initio calculations. These calcnlations represent a treatment of electron distribution and electron motion which implies that individual electrons are one-electron functions containing a product of spatial functions called molecular orbitals hi(x,y,z), 4/2(3 ,y,z), and so on. In the simplest version of this theory, a single assignment of electrons to orbitals is made. In turn, the orbitals form a many-electron wave function, 4/, which is the simplest molecular orbital approximation to solve Schrodinger s equation. In practice, the molecular orbitals, 4 1, 4/2,- -are taken as a linear combination of N known one-electron functions 4>i(x,y,z), 4>2(3,y,z) ... 

The electronic Schrodinger equation is still intractable and further approximations are required. The most obvious is to insist that electrons move independently of each other. In practice, individual electrons are confined to functions termed molecular orbitals, each of which is determined by assuming that the electron is moving within an average field of all the other electrons. The total wavefunction is written in the form of a single determinant (a so-called Slater determinant). This means that it is antisymmetric upon interchange of electron coordinates. ... 

Molecular Orbital Models. Methods based on writing the many-electron solution of the Electronic Schrodinger Equation in terms of a product of one-electron solutions (Molecular Orbitals). 

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